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Modeling the Asymmetry of Stock Movements Using Price Ranges. Ray Y. Chou Academia Sinica “ The 2002 NTU International Finance Conference” Taipei. May 24-25, 2002. Motivation. Provide separate dynamic models for the upward-range and the downward-range to allow for asymmetries. - PowerPoint PPT Presentation
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Modeling the Asymmetry of Stock Movements Using Price
Ranges
Ray Y. ChouAcademia Sinica
“ The 2002 NTU International Finance Conference” Taipei. May 24-25, 2002
Motivation
• Provide separate dynamic models for the upward-range and the downward-range to allow for asymmetries.
• Factors driving the upward movements and the downward movements maybe different.
• Upward range applications: market rallies, call options, historical new highs, limit order to sell
• Downward range applications: Value-at-Risk, put options, limit order to buy
Main Results• ACARR is similar to CARR and ACD but
with a different limiting distribution and with new interpretations and implications.
• Properties: QMLE, Distribution• Empirical results using daily S&P500 index
show asymmetry in dynamics, leverage effect, periodic patterns and interactions of upward and downward movements.
• Volatility forecast accuracy: ACARR>CARR>GARCH
Range as a measure of the “realized volatility”
• Simpler and more natural than the sum-squared-returns (measuring the integrated volatility) of Anderson et.al.(2000)
• Parkinson (1980) and others have established the efficiency gain of range over standard method in estimating volatilities
• Chou (2001) proposed CARR, a dynamic model for range with satisfactory performance
Discrete sampling from a continuous process
Let Pt be the logarithmic price of a speculative asset observed at
time t, t =1,2,…T. Pt are taken to be realizations of a price
process {P}, which is assumed to be a continuous process.
Let OPENtP , CLOSE
tP , HIGHtP , LOW
tP be the opening, closing, high and
low prices, in natural logarithm, between t-1 and t.
Upward range and downward range
Define UPRt and DWNRt as the differences between
the daily highs, daily lows and the opening price
respectively, at time t, i.e.,
OPENt
HIGHtt PPUPR
).( OPENt
LOWtt PPDWNR
Range and one-sided ranges
Note that these two one-sided ranges, UPR t and
DWNR t, represent the maximum returns and the
min imum returns (in percentage size) respectively
over the unit time interval [t-1, t]. Further, the
range R t, is defined to be
LO W
tH IG H
tt PPR , or
ttt DWNRUPRR .
The Conditional Autoregressive Range Expectation (CARR) model in Chou
(2001)
tttR
jt
q
jjit
p
iit R
11
t ~ iid f(.)
The Asymmetric Conditional Autoregressive Range Expectation -ACARR(p,q) model
ut
uttUPR
dt
dttDWNR
u
jt
q
j
ujit
p
i
ui
uut UPR
11
d
jt
q
j
djit
p
i
di
ddt DWNR
11
Distribution assumptions in ACARR
ut ~ iid fu(.), d
t ~ iid fd(.)
It is shown that the limiting distributions fu(.) and fd(.) are identical but are different from the error distribution for the CARE model in Chou (2000b).
ACARRX(p,q) – ACARR(p,q) with exogenous variables
Let be some variables measurable at time t-1, the ACAREX(p,q) model is defined to be
,1111
ltl
L
l
u
jt
q
j
ujit
p
i
ui
uut XUPR
,1111
ltl
L
l
d
jt
q
j
djit
p
i
di
ddt XDWNR
Explanatory variables in the ACARRX(p,q) model
• Lagged returns – leverage effect• Periodic (weekday) pattern• Transaction volumes• Interaction tems – lagged DWNR in expecte
d UPR and lagged UPR in expected DWNR
Properties of ACARR
• Same as ACD of Engle and Russell (1998) but with a known limiting distribution for the error term
• A conditional mean model • An asymmetric model for volatilities
Sources of asymmetry for an ACARRX(1,1) model
– short term shock impact – long term persistence of shocks – speed of mean-reverting ‘s – effects of leverage, periodic pattern,
interaction terms, among others
A special case of ACARR: Exponential ACARR(1,1) or EACARR(1,1)
• It’s useful to consider the exponential case for f(.), the distribution of the normalized range or the disturbance.
• Like GARCH models, a simple (p=1, q=1) specification works for many empirical examples.
ACARR vs. ACD identical formula
• ACARR• Range data, positive
valued, with fixed sample interval
• QMLE with EACARR• Known limiting
distribution• A new volatility
model
• ACD• Duration data, positive
valued, with non-fixed sample interval
• QMLE with EACD• Unknown limiting
distribution• Hazard rate
interpretation
The QMLE property
• Assuming any general density function f(.) for the disturbance term t, the parameters in ACARR can be estimated consistently by estimating an exponential-ACARR model.
• Proof: see Engle and Russell (1998), p.1135
The QMLE Estimation
• Consistent standard errors are obtained by employing the robust covariance method in Bollerslev and Wooldridge (1987).
• See Engle and Russell (1998).
Empirical example: S&P500 daily index
• Sample period: 1962/01/03 – 2000/08/25• Data source: Yahoo.com• Models used: EACARR(1,1), EACARRX(p,q)• Both daily and weekly observations are used
for estimation• Forecast comparison of CARR and ACARR
-30
-20
-10
0
10
2000 4000 6000 8000
UPR DWNR
Figure 1: Daily UPR and Daily DWNR, S&P500, 1962/1-2000/8
Series: MAXSample 1 9700Observations 9700
Mean 0.737026Median 0.635628Maximum 9.052741Minimum 0.000000Std. Dev. 0.621222Skewness 2.110659Kurtosis 15.76462
Jarque-Bera 73055.13Probability 0.000000
0
400
800
1200
1600
2000
2400
0.00 1.25 2.50 3.75 5.00 6.25 7.50 8.75
Series: MAXSample 1 9700Observations 9700
Mean 0.737026Median 0.635628Maximum 9.052741Minimum 0.000000Std. Dev. 0.621222Skewness 2.110659Kurtosis 15.76462
Jarque-Bera 73055.13Probability 0.000000
Figure 2: Daily UPR of S&P500 Index, 1962/1-2000/8
Series: AMINSample 1 9700Observations 9700
Mean 0.727247Median 0.598360Maximum 22.90417Minimum 0.000000Std. Dev. 0.681298Skewness 5.456951Kurtosis 128.1277
Jarque-Bera 6376152.Probability 0.000000
0
1000
2000
3000
4000
5000
6000
7000
8000
0 2 4 6 8 10 12 14 16 18 20 22
Series: AMINSample 1 9700Observations 9700
Mean 0.727247Median 0.598360Maximum 22.90417Minimum 0.000000Std. Dev. 0.681298Skewness 5.456951Kurtosis 128.1277
Jarque-Bera 6376152.Probability 0.000000
Figure 3: Daily DWNR of S&P500 Index, Unsigned, 1962/1-2000/8
Series: AMINSample 1 9700Observations 9699
Mean 0.724960Median 0.598329Maximum 8.814014Minimum 0.000000Std. Dev. 0.643037Skewness 2.271508Kurtosis 15.61285
Jarque-Bera 72630.55Probability 0.000000
0
400
800
1200
1600
2000
2400
0.00 1.25 2.50 3.75 5.00 6.25 7.50 8.75
Series: AMINSample 1 9700Observations 9699
Mean 0.724960Median 0.598329Maximum 8.814014Minimum 0.000000Std. Dev. 0.643037Skewness 2.271508Kurtosis 15.61285
Jarque-Bera 72630.55Probability 0.000000
Figure 4: Daily DWNR w/o Crash, Unsigned, 1962/1-2000/8
Nobs Mean Median Max Min Std Dev 1 12 Q(12)
Full sampleRANGE 1/2/62-8/25/00 9700 1.464 1.407 22.904 0.145 0.76 0.629 0.575 0.443 30874UPR 1/2/62-8/25/00 9700 0.737 0.636 9.053 0 0.621 0.308 0.147 0.172 3631DWNR 1/2/62-8/25/00 9700 -0.727 -0.598 0 -22.9 0.681 0.326 0.181 0.162 4320
Before structural shiftRANGE 1/2/62-4/20/82 5061 1.753 1.643 9.326 0.53 0.565 0.723 0.654 0.554 21802UPR 1/2/62-4/19/82 5061 0.889 0.798 8.631 0 0.581 0.335 0.087 0.106 1199DWNR 1/2/62-4/19/82 5061 -0.864 -0.748 0 -6.514 0.559 0.378 0.136 0.163 2427
After structural shiftRANGE 4/21/82-8/25/00 4639 1.15 0.962 22.904 0.146 0.818 0.476 0.414 0.229 11874UPR 4/21/82-8/25/00 4639 0.572 0.404 9.053 0 0.622 0.189 0.089 0.125 651DWNR 4/21/82-8/25/00 4639 -0.578 -0.388 0 -22.9 0.767 0.247 0.147 0.101 994
Table 1: Summary Statistics of the Daily Range, Upward Range and Downward Range of S&P500 Index, 1/2/1962-8/25/2000
0.05
0.10
0.15
0.20
0.25
0.30
0.35
20 40 60 80 100 120 140 160 180 200
RHO_UPR RHO_DWNR RHO_DWNR(w/o crash)
Figure 5: Correlograms of Daily UPR and DWNR
T ab le 2 : Q M L E E st im a tion of AC A RR Using D a ily Upw ar d R an ge of S& P 5 00 Ind ex 1/2 /1 96 2-8 /25 /2 00 0
t ~ iid f( .) E s tima tio n i s c a rr ied ou t us in g th e QM L E m e th od h e nc e it 's equ iva lent to e stim a t in g a n E xp on en tia l AC AR R( X ) (p, q) or a nd E AC AR R (X ) (p ,q) m o de l . N u mbe rs in p a re nthe se s a re t-ra tios ( p-v a lu e s) w ith ro bu st sta n da rd e rr ors fo r the m od el c o ef fi cie nts ( Q sta tist ic s). L L F is th e log l ike l ih oo d fun c tion . A C AR R( 1,1 ) AC A R R( 2,1 ) A CA RR X ( 2,1 )-a A CA RR X ( 2,1 )-b A CA RR X ( 2,1 )-c
L L F -1 20 35 .2 0 -1 20 11 .8 6 -1 19 55 .7 8 -1 19 49 .6 4 -1 19 50 .3 2
c on sta n t 0. 00 2 [ 3.2 16 ] 0. 00 1 [ 3.1 45 ] -0. 00 2 [-0.6 10 ] -0. 00 3 [-0.5 51 ] -0. 00 4 [-0.9 73 ]
UP R( t-1 ) 0 .0 3 [ 8.8 73 ] 0. 14 5 [1 0.8 37 ] 0. 20 3 [1 4.0 30 ] 0. 17 9 [1 1.8 56 ] 0. 18 6 [1 2.8 45 ]
UP R( t-2 ) -0. 12 6 [-9.1 98 ] -0. 11 7 [-9 ..4 48 ] -0. 11 2 [-8.8 79 ] -0. 11 5 [-9.2 45 ]
( t-1 ) 0. 96 8 [ 26 7.9 93 ] 0. 97 8 [ 34 1.9 23 ] 0. 90 3 [ 69.6 43 ] 0. 87 1 [4 8.4 26 ] 0. 87 7 [5 2.9 42 ]
r( t-1 ) -0. 05 7 [-8.4 31 ] -0. 01 8 [-1.9 59 ] -0. 02 3 [-2.7 34 ]
T U E 0. 05 8 [ 3.4 23 ] 0. 05 9 [ 3.4 75 ] 0. 05 9 [ 3.4 81 ]
W E D 0 .0 2 [ 1.2 71 ]
SD 0 .0 00 0 [ 0.2 01 ] -0. 00 3 [-1 .6 47 ]
DW NR (-1 ) 0. 04 6 [ 4.8 68 ] 0. 04 2 [ 4.8 03 ]
Q (12 ) 18 4. 4 [ 0.0 00 ] 2 2. 34 6 [ 0.0 34 ] 2 2. 30 4 [ 0.0 34 ] 2 0. 28 2 [ 0.0 62 ] 2 0. 50 3 [ 0.0 53 ]
tttU P R
,1111
ltl
L
l
u
jt
q
j
ujit
p
i
ui
uut XU P R
T able 3 : Q M L E Es tim at io n o f ACA RR U sin g D aily Do wnw ard Ra ng e of S& P5 00 In dex 1/2 /1 9 62-8 /25 /2 0 00
t ~ iid f(.) Estimatio n is carried ou t usin g th e QM LE m ethod h en ce it's equ ivalent to es tim atin g an Expo n en tial AC A RR (X )(p,q) or and EA CA R R(X)(p ,q) mo del. N um bers in p arentheses ar e t-ratio s(p-values) with rob ust standard er ro rs for the m od el co eff icients ( Q statistics). LLF is the log lik elih oo d fun ctio n. A C AR R(1,1) A C AR R(2,1) AC A RR X(2,1)-a AC A RR X(2,1)-b AC A RR X(2,1)-c
LLF -11 929.39 -11 889.61 -11 87 3.54 -11 86 8.55 -11 870.14
con stan t 0 .0 14 [5.905] 0 .0 04 [4.088] 0 .017 [4.37 3] 0 .0 17 [3.23 5] 0 .0 17 [4.417]
DW N R (t-1) 0 .0 84 [11.83 4] 0 .2 29 [1 6.277] 0 .252 [1 6.123] 0 .2 33 [1 4.77 0] 0 .2 39 [1 6.36 4]
DW N R (t-2) -0 .1 95 [-1 3.48 9] -0 .189 [-1 2.811] -0 .1 85 [-1 2.59 4] -0 .1 86 [-12.897]
t-1 0 .8 97 [1 01 .0 2] 0 .9 61 [1 99 .02] 0 .927 [8 7.639] 0 .9 06 [6 1.21 2] 0 .9 11 [6 3.89 3]
r (t-1) 0 .023 [4.72 1] -0 .0 09 [-1.187]
TU E -0 .0 08 [0.50 3]
W ED -0 .051 [-3.58 2] -0 .0 53 [-3.617] -0 .0 52 [-3.58 7]
SD -0 .002 [-2.12 4] 0 .0 01 [0.90 4]
UP R(-1) 0 .0 37 [4.16 4] 0 .0 28 [5.084]
Q (1 2) 192 .8 [0.000] 1 8.94 [0.009] 22 .2 27 [0.03 5] 14 .4 22 [0.27 5] 14 .7 74 [0.254]
tttD W N R ltl
L
l
d
jt
q
j
djit
p
i
di
ddt XD W N R 1
111
-10
-8
-6
-4
-2
0
2
4
2000 4000 6000 8000
UPR_NEG LAMBDA_UPR
Figure 6: Expected and Observ ed Daily UPR, 1962/1-2000/8
-25
-20
-15
-10
-5
0
5
10
2000 4000 6000 8000
DWNR LAMBDA_DWNR
Figure 7: Expected and Observ ed Daily DWNR, 1962/1-2000/8
Series: ET_MAXSample 1 9700Observations 9700
Mean 1.233526Median 1.040290Maximum 29.61918Minimum 0.000000Std. Dev. 1.093848Skewness 3.854160Kurtosis 59.76949
Jarque-Bera 1326553.Probability 0.000000
0
1000
2000
3000
4000
5000
0 5 10 15 20 25 30
Series: ET_MAXSample 1 9700Observations 9700
Mean 1.233526Median 1.040290Maximum 29.61918Minimum 0.000000Std. Dev. 1.093848Skewness 3.854160Kurtosis 59.76949
Jarque-Bera 1326553.Probability 0.000000
Figure 8: Histogram of Daily et_UPR, 1962/1-2000/8
Series: ET_MINSample 1 9700Observations 9700
Mean 0.837894Median 0.735879Maximum 11.63559Minimum 0.000000Std. Dev. 0.688907Skewness 2.306857Kurtosis 18.96869
Jarque-Bera 111665.3Probability 0.000000
0
500
1000
1500
2000
2500
3000
3500
0 2 4 6 8 10 12
Series: ET_MINSample 1 9700Observations 9700
Mean 0.837894Median 0.735879Maximum 11.63559Minimum 0.000000Std. Dev. 0.688907Skewness 2.306857Kurtosis 18.96869
Jarque-Bera 111665.3Probability 0.000000
Figure 9: Histogram of daily et_DWNR, 1962/1-2000/8
0
2
4
6
8
10
0 2 4 6 8 10
ET_UPR
Expo
nent
ial Q
uant
ile
Figure 10: Q-Q Plot of et_UPR
0
2
4
6
8
1 0
0 2 4 6 8 1 0
E T _ D W N R
Exp
onen
tial
Qua
ntil
eF ig u re 1 1 : Q -Q p lo t o f e t-D W N R
Figure 11: Q-Q plot of et-DWNR
Ta b le 4: A C A R R v e rsu s C A R R
In -s am p le V ola ti l ity For ec a st Co m pa r iso n U sin g T hre e M ea su re d V ola ti l it ie s as Be nc h m a rks. T he thre e m e a sur e s of vo la t il i ty a re R N G , R ET S Q a nd A R E T : re spe c tiv ely,
da i ly ra n ge s, sq ua re d -da i ly -re turn s, a n d a bso ul te da i ly- r et u rn. A CA R R (1 ,1) m od e l i s f it t ed fo r th e ra ng e ser ie s a n d a A CA RR m o de l s a re fi t te d fo r the u pw ar d r an ge
an d th e do w nw a rd ra ng e ser ie s. F V (C A R R) (F V (A CA R R )) is th e for e ca st e d v ola ti li t y u s ing CA RR ( A CA RR ). ( F V ( A CA RR )) i s th e fo re ca ste d ra ng e us ing t he su m o f the f orc a st ed u pw a rd r an ge a nd d ow n w a rd ra n ge .
P ro pe r tr an sfo rm a t io ns a r e m ad e fo r a d ju sting the d if fe re nc e b etw e e n a va ria nc e e s tim a to r an d a s ta n da rd -de v ia t io n e st im ator . N u m be rs in p ar en the se s a re t-ra tios .
M V t = a + b F V t(C A R R) + ut M V t = a + c F V t(A C A R R) + ut M V t = a + b F V t(C A R R) + c F V t(A C A R R) + ut
M e as ure d V o la t il i ty E x plan a to ry V a ria ble s c o nsta nt FV (C A R R) FV ( A C A R R) A d j . R -sq. S .E .
R N G -0. 06 7 [-0 .36 6] 1 .0 05 [9 6.2 9] 0 .4 89 0 .5 43
R N G -0 .0 06 [-4 .14 8] 1 .0 47 [ 10 1.0 2] 0 .5 13 0 .5 31
R N G -0 .0 67 [0 .63 2] 0 .0 21 [ 0.4 6] 1 .0 26 [2 1.8 3] 0 .5 13 0 .5 31
RE T SQ -1 .2 03 [-1.3 5] 0 .3 97 [1 4.2 5] 0. 02 5 .7 25
RE T SQ -0 .2 65 [-2.9 4] 0 .4 59 [1 6.0 2] 0 .0 26 5 .7 09
RE T SQ -0 .2 49 [-2.7 6] -0 .1 91 [-2.3 2] 0 .6 44 [ 7.6 1] 0 .0 26 5 .7 08
A R E T 1 .1 42 [ 7.4 1] 0 .3 34 [2 7.0 7] 0. 07 0 .6 42
A R E T 0 .1 13 [ 5.8 5] 0 .3 54 [2 8.2 8] 0 .0 76 0 .6 39
A R E T 0 .1 15 [ 5.9 5] -0 .1 06 [-1.9 1] 0 .4 58 [ 8.0 9] 0 .0 76 0 .6 39
Extensions
• Robust ACARR – Interquartile range• Multivariate ACARR• Nonparametric or semiparametric ACARR• Other data sets and simulations• Long memory ACARR’s – IACARR, FIAC
ARR,…• ACARR and option price models
Conclusion
• ACARR is effective in modeling upward and downward market movements.
• Asymmetry found: dynamics, leverage effect, periodic patterns, interaction terms
• CARR provides more accurate volatility forecasts than GARCH (Chou (2001)) and ACARR gives further improvements.